Question

Consider a political discussion group consisting of 4 Democrats, 4 Republicans, and 2 Independents. Suppose that two group members are randomly selected, insuccession, to attend a political convention. Find the probability of selecting no Independents(Type an integer or a simplified fraction)

Answer 1

Let D, R, and I, denote the set of politicians being Democrats, Republicans, and Independents.

According to the given problem,

$\begin{gathered} n(D)=4 \\ n(R)=4 \\ n(I)=2 \end{gathered}$

Consider that the probability of an event is given by,

$\text{Probability}=\frac{\text{ Number of favourable outcomes}}{\text{ Total number of outcomes}}$

As per the given problem, the favourable event is that the two selected politicians at succession are not Independents.

The number of ways of selecting 2 politicians such that both of them are Independents,

$\begin{gathered} =^2C_2 \\ =\frac{2!}{2!\cdot(2-2)!} \\ =\frac{1}{0!} \\ =1 \end{gathered}$

So there is only 1 favourable outcome.

The total number of ways of selecting 2 politicians from the group is,

$\begin{gathered} =^{10}C_2 \\ =\frac{10!}{2!\cdot(10-2)!} \\ =\frac{10\cdot9\cdot8!}{(2\cdot1)\cdot8!} \\ =5\cdot9 \\ =45 \end{gathered}$

Then the corresponding probability is given by,